where \(K\)  quantifies the noise of imitation process [1]}. Importantly, players can also adopt an attitude from each other where they use the same imitation probability defined by Eq. REF . These processes, however, are independent from each others, otherwise we may observe artificial consequence of coupled imitation processes. At the beginning of the Monte Carlo (MC) simulations we distribute both \(C\) , \(D\)  strategies and \(``A"\) , \(``O"\)  attitudes randomly among players who are on a square grid. During an elementary process we randomly select a target player \(i\)  and a neighboring player \(j\) . When the corresponding payoff values are calculated then we determine the imitation probability defined by Eq. REF . After target player adopts the strategy and/or attitude of source player with the imitation probability. This step is repeated \(L \times L\)  times, which establishes a MC step. In our simulations we typically applied \(L=600\)  system size and 50000 MC steps to reach the stationary state, but in the vicinity of transition points we checked our data to avoid finite-size problems. To obtain comparable results with previous studies [2]}, [3]} we used \(K=0.5\)  noise level, but we have checked other values to confirm the robustness of our observations.
